Introduction to Advanced Mathematics, MATH 215, Fall 2017

Instructor: Daniel Groves, 727 SEO e.mail

Course webpage: http://www.math.uic.edu/~groves/teaching/2017-18/215/

Course hours:

MWF, 11:00-11:50AM, Room 310, Addams Hall.

Office hours:

Mondays 10am, Wednesdays 12pm.

(Or, you can make an apointment by e.mail or try stopping past my office.)

Text: "An Introduction to Mathematical Reasoning", by P. Eccles, Cambridge University Press. ISBN:9780521597180
(I do not require you to buy this book, but you might find it useful.)

Course description:
The goal of this course is to learn how to create and write mathematical proofs, and to learn why one might want to do such a thing. We will introduce and study some important mathematical concepts used in advanced mathematics courses, particularly equivalence relations.

Assessment:
There will be homework for most classes, two in class midterm exams and a final exam. Since there will be a lot of writing, explaining and critiquing in class, there will also be a class participation component of the grade. The relative weighting of these components will be:
• Homework: 20%
• Class participation: 10%
• Midterm exams: 20% each
• Final exam: 30%

Exams:

Two midterms, and one final, dates and times to be announced.

• The first midterm will be Monday, October 9, in class.

Here are some worked solutions for Midterm 1.

• The second midterm will be Wednesday, November 15, in class.

Here are some worked solutions for Midterm 2.

Daily Homework:

• For Wednesday, August 30, at the beginning of class. In class on Monday 8/28, we will work on this worksheet. For the beginning of class on Wednesday, prove Proposition 4. Decide whether or not Conjecture 5 is true. If Conjecture 5 is true, prove it. If it is not true, prove it is not true.

• For Friday, September 1, at the beginning of class. Prove Propositions 6 and 7 from the above worksheet. If you did not do the Conjecture 5 HW from Wednesday, do this as well.

• For Wednesday, September 6, at the beginning of class. Prove Propositions 8 and 9 from Worksheet 1.

Here is the second worksheet, which we will beginning working on on Wednesday.

• For Friday, September 8, at the beginning of class. Prove Propositions 10, 11, 12 and 13 from Worksheet 2.

• For Wednesday, September 13, at the beginning of class. Do this homework. This homework will be graded (given a score out of 25), and the points will count towards your homework score. [Other homework which is not grade but just read and commented on counts as well, but only for having done it or not.]

Here is the third worksheet, which we will begin working on on either September 13 or 15.

• For Friday, September 15, at the beginning of class. Prove Propositions 16, 17 and 18 from Worksheet 2.

• For Monday, September 18, at the beginning of class. Prove Propositions 22 and 23.

• For Wednesday, September 20, at the beginning of class. In class on Monday, you were given somebody else's homework to grade. Do this grading and come ready to discuss it with that person.

• For Friday, September 22, at the beginning of class. Prove Lemma 30 and Theorem 31.

• For Monday, September 25, at the beginning of class. Prove Lemma 30 and Theorem 31. (Again.)

• Here is a worksheet on sets, relations, and equivalence relations.

• For Monday October 2, at the beginning of class. Do this homework. It will be graded out of 25 points. Here are some worked solutions for this homework.

Here is the fourth worksheet on number theory.

• For Friday, September 29, at the beginning of class. Prove statements 1,2 and 3 of Theorem 33, from Number Theory IV.

• By popular(?) request, here is a proof of Theorem 31.

• For Friday, October 13, at the beginning of class. Prove statements 6, 7 and 8 from Theorem 33. Also Prove Proposition 34, Theorem 37 and Corollary 38.

• Here is the next worksheet, which we will begin working on on Friday 10/13.

• For Monday, October 16, at the beginning of class. The homework was explained in class. See also here.

• For Wednesday, October 18, at the beginning of class. Prove Propositions 3 and 4 from the Fields Worksheet.

• For Friday, October 20, at the beginning of class. Let n be a natural number (at least 2) and let S be the set of equivalence classes of integers mod n. Define addition and multiplication as we did in class (see the equivalence relations worksheet). Prove that with these operations S satisfies axioms (A1), (A2) and (A3) from the Fields worksheet.

• For Monday, October 30, at the beginning of class. Do this homework. If will be graded out of 25 points. Here are some worked solutions to this homework.

• For Wednesday, October 25, at the beginning of class. Prove Propositions 5,6,7 and 8 from the Fields Worksheet.

• Here is the next number theory worksheet, which we will start working on on 10/25.

• Here is the next number theory worksheet, which we will start working on on 11/3.

• For Wednesday, November 8, at the beginnig of class. Prove Proposition 54 and Theorem 55.

• For Friday, November 10, at the beginning of class. Prove the following lemma:

Lemma. Let n be a natural number and p be a prime. If n | p then either n =1 or n = p.

Then prove Proposition 54 and Theorem 55 again.

Here is the next worksheet (on induction) which we will start working on on November 17.

NO CLASS ON WEDNESDAY, November 22.

• For Monday, November 27, at the beginning of class. Do Exercise 6 from the worksheet on induction (unless you already turned it in).

• For Wednesday, November 29, at the beginning of class. Prove Propositions 9 and 10 from the Induction worksheet.

• For Monday, December 4, at the beginning of class. Prove Proposition 14 from the Induction worksheet.

Here is the next (and probably last) worksheet, which we will work on from Monday, December 4.